Geometric structures on complex manifolds: talks and abstracts
Home | Venue | Schedule | Program | Poster | PDF
Dmitri Alekseevsky (Masaryk University in Brno)
Lorentzian manifolds with large isometry group
I give a survey of results about isometry group of Lorentzian manifolds
and will describe some classes of homogeneous Lorentzian manifolds
including homogeneous manifolds of a semisimple Lie group and
manifolds with weakly irreducible isotropy group.
Semyon Alesker (Tel Aviv University)
Quaternionic Monge-Ampere equations and HKT-geometry.
A notion of quaternionic Monge-Ampere equation will be introduced.
These are non-linear second order elliptic equations which make sense on so
called hypercomplex manifolds, in
particular on the flat quaternionic space.
They admit an interpretation in the framework of Hyper Kahler with Torsion
(HKT) geometry (to be explained in the talk).
We formulate a quaternionic version of the Calabi conjecture, and state a
number of partial results towards
its proof. Part of the results are joint with M. Verbitsky.
An isometric action of a connected Lie group $H$ on a
Riemannian manifold $M$ is called polar if there exists a
connected closed submanifold $\Sigma$ of $M$ such that
$\Sigma$ meets each orbit of the action and intersects it
orthogonally. An elementary example of a polar action
comes from the standard representation of $SO_n$ on
${\mathbb R}^n$. Further examples of polar actions can be
constructed from Riemannian symmetric spaces. Let $M =
G/K$ be a Riemannian symmetric space and denote by $o$ a
fixed point of the $K$-action on $M$. Then the isotropy
representation $\pi : K \to O(T_oM)$ of $K$ on the tangent
space $T_oM$ of $M$ at $o$ induces a polar action. Dadok
established in 1985 a remarkable, and mysterious, relation
between polar actions on Euclidean spaces and Riemannian
symmetric spaces. He proved that for every polar action on
${\mathbb R}^n$ there exists a Riemannian symmetric space
$M = G/K$ with $\dim M = n$ such that the orbits of the
action on ${\mathbb R}^n$ and the orbits of the $K$-action
on $T_oM$ are the same via a suitable isomorphism
${\mathbb R}^n \to T_oM$.
Soon afterwards an attempt was made to classify polar
actions on symmetric spaces. For irreducible symmetric
spaces of compact type the final step for a complete
classification appears to have been just completed by
Kollross using yet unpublished work of Lytchak on polar
foliations. In the talk I want to focus on symmetric
spaces of noncompact type. For actions of reductive groups
one can use the concept of duality between symmetric
spaces of compact type and of noncompact type. However,
new examples and phenomena arise from the geometry induced
by actions of parabolic subgroups, for which there is no
analogon in the compact case. I plan to discuss the main
difficulties one encounters here and some partial
solutions. The only complete classification known so far
has just been obtained in joint work with Jos\'{e} Carlos
D\'{i}az-Ramos for the complex hyperbolic plane.
Roger Bielawski (University of Leeds)
Hypercomplex and pluricomplex geometry
I'll will describe a new type of geometric structure on complex
manifolds. It can be viewed as a deformation of hypercomplex structure, but
it also leads to a special type of hypercomplex geometry. These structures
have both algebro-geometric and differential-geometric descriptions, and there
are interesting examples arising from physics.
Gil Cavalcanti (Utrecht University)
Generalized Kahler structures on moduli space of instantons
We show how the reduction procedure for generalized Kahler
structures can be used to recover Hitchin's results about the
existence of a generalized Kahler structure on the moduli space of
instantons on bundle over a generalized Kahler manifold. In this setup
the proof follows closely the proof of the same claim for the Kahler
case and clarifies some of the stranger considerations from Hitchin's
proof.
Vicente Cortés (Hamburg University)
From cubic polynomials to complete quaternionic Kahler manifolds
I will explain two supergravity constructions, which allow to construct
certain special Riemannian manifolds starting with other special
Riemannian manifolds. We show that the resulting manifolds are complete
if the original manifolds are. By composition of the two constructions
we obtain complete quaternionic Kahler manifolds out of certain
cubic hypersurfaces. At the end I will formulate two open problems
concerning such hypersurfaces.
The talk is based on
arXiv:1101.5103 (hepth, mathdg).
Isabel Dotti (University of Cordoba, Argentina)
Some restrictions on existence of abelian complex
structures
We describe the structure of Lie groups admitting left
invariant abelian complex structures in terms of
commutative associative algebras. More precisely, we
consider a distinguished class of Lie algebras
admitting abelian complex structures given by abelian
double products. The structure of these Lie algebras
can be described in terms of a pair of commutative
associative algebras satisfying a compatibility
condition. We will show that when g is a Lie algebra
with an abelian complex structure J, and g decomposes
as g = u + Ju, with u an abelian subalgebra, then g is an
abelian double product.
Joint work with A. Andrada and M. L. Barberis
In this talk I wil introduce a new equation on the compact Kahler
manifolds. Solution of this equation corresponds to the Calabi--Yau
metric. New equation describes deformation of complex structure,
while Monge--Ampere equation describes deformation of symplectic structure.
Anna Fino (Torino University)
Special Hermitian structures and symplectic geometry
Symplectic forms taming complex structures on
compact manifolds are strictly related to a special type
of Hermitian metrics, known in the literature as "strong
Kaehler with torsion" metrics. I will present general
results on "strong Kahler with torsion" metrics, their
link with symplectic geometry and more in general with
generalized complex gometry. Moreover, I will show
for certain 4-dimensional non-Kaehler symplectic
4-manifolds some recent results about the Calabi-Yau
equation in the context of symplectic geometry.
Akira Fujiki (Osaka)
Joyce twistor space and the associated K\"ahler class
Fix a smooth action of a real two-torus on the connected
sum of m copies of
complex projective plane. Then the invariant self-dual structures
constructed by Joyce,
and hence the associated twistor spaces, depend on real (m-1)-dimensional
parameter.
We can then associate each twistor space a Kaehler class on a fixed open
rational surface,
which makes the moduli space of Joyce twistor spaces a domain of the
projectified K\"ahler
cone of the surface. This result then gives a nice description of the
families of anti-self-dual
bihermitian structures on hyperbolic Inoue surfaces constructed previously
with Pontecorvo.
The aim of this lecture is to show that, apart from the complex
Grassmannians $Gr_2(C^{n+2})$, the compact quaternionic quaternion-Kaehler
manifolds of positive type admit no almost complex structure, even in the
weak sense (joint work with Andrei Moroianu and Uwe Semmelmann).
Ryushi Goto (Osaka)
Deformations of L.C.K. structures and generalized Kaehler
structures
A notion of geometric structures which includes
locally conformally Kaehler and generalized Kaehler structures will
be introduced.
Unobstructed deformations of the structures are discussed from the
unified view point.
Deformations on non-Kaehler manifolds such as Vaisman manifolds and
Inoue surfaces
will be given.
We describe a family of calibrations arising naturally on
a hyperkaehler manifold M. These calibrationscalibrate the
holomorphic Lagrangian, holomorphic isotropic and
holomorphic coisotropic subvarieties. When M is an HKT
(hyperkahler with torsion) manifold with holonomy
SL(n,H), we construct another family of calibrations, which
calibrate holomorphic Lagrangian and holomorphic
coisotropic subvarieties. They are (generally speaking)
not parallel with respect to any torsionless connection on
M. We note also that there are examples of complex
isotropic submanifolds in SL(n, H) manifolds with HKT
structure, which can not be calibrated by any form, unlike
the Kaehler case.
A homogeneous Hermitian manifold M with its homogeneous Hermitian
structure h, defining a locally conformally Kaehler structure w is
called a homogeneous locally conformally Kaehler or shortly a homogeneous
l.c.K.
manifold. If a simply connected homogeneous l.c.K. manifold M=G/H,
where G is a connected Lie group and H a closed subgroup of G,
admits a free action of a discrete subgroup D of G from the left,
then a double coset space D\G/H is called a locally homogeneous
l.c.K. manifold. We discuss explicitly homogeneous and locally
homogeneous l.c.K. structures on Hopf surfaces and Inoue surfaces,
and their deformations. We also classify all complex surfaces
admitting locally homogeneous l.c.K. structures.
We show as a main result a structure theorem of compact homogeneous
l.c.K. manifolds, asserting that it has a structure of a holomorphic
principal fiber bundle over a flag manifold with fiber a 1-dimensional
complex torus. As an application of the theorem, we see that only
compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf
surfaces of homogeneous type. We also see that there exist no compact
complex homogeneous l.c.K. manifolds; in particular neither complex
Lie groups nor complex paralellizable manifolds admit their compatible
l.c.K. structures.
We show as a main result a structure theorem of compact homogeneous
l.c.K. manifolds, asserting that it has a structure of a holomorphic
principal fiber bundle over a flag manifold with fiber a 1-dimensional
complex torus. As an application of the theorem, we see that only
compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf
surfaces of homogeneous type. We also see that there exist no compact
complex homogeneous l.c.K. manifolds; in particular neither complex
Lie groups nor complex paralellizable manifolds admit their compatible
l.c.K. structures.
This talk is based on a joint work with Y. Kamishima
"Locally conformally Kaehler structures on homogeneous spaces"
(arXiv:1101.3693).
Stefan Ivanov (Sofia University)
Extremals for the Sobolev-Folland-Stein inequality, the quaternionic
contact Yamabe problem and related geometric structures
We describe explicitly non-negative extremals for the Sobolev inequality
on the quaternionic Heisenberg groups and determine the best constant
in the $L^2$ Folland-Stein embedding theorem involving quaternionic
contact (qc) geometry and the qc Yamabe equation. Translating the problem
to the 3-Sasakian sphere, we determine the qc Yamabe invariant on the
spheres. We describe explicitly all solutions to the qc Yamabe equation
on the seven dimensional quaternionic Heisenberg group. The main tool is
the notion of qc structure and the Biquard connection. We define a
curvature-type tensor invariant called qc conformal curvature in terms
of the curvature and torsion of the Biquard connection and show that a
qc manifold is locally qc conformal (gauge equivalent) to the standard flat
qc structure on the Heisenberg group, or equivalently, to the 3-sasakian
sphere if and only if the qc conformal curvature vanishes. Possibly, this
will help to reduce the qc Yamabe problem to that of the spherical qc manifolds.
Julien Keller (Aix Marseille University)
Chow stability and the projectivisation of stable bundles
We will discuss the Chow stability of the projectivisation of
a Gieseker stable bundles over a surface endowed with a constant
scalar curvature Kahler metric. We will provide an example of a
smooth manifold which is Chow stable but not asymptotically Chow
stable. This is a joint work with J. Ross.
Dario Martelli (King's College, London)
Geometric structures arising in string theory
Several interesting differential-geometric structures
arise in the context of string theory from requiring
supersymmetry. In particular, this implies the existence
of spinor fields obeying certain differential equations. I
will discuss the physical motivations behind these
structures and I will review examples of explicit
constructions, including: special holonomy manifolds,
complex non-Kaehler manifolds, and Sasaki-Einstein manifolds.
Ruxandra Moraru (Waterloo University)
Compact moduli spaces of stable bundles on Kodaira surfaces
Abstract: In this talk, I will examine the geometry of moduli spaces of
stable bundles on Kodaira surfaces, which are non-Kaehler compact surfaces
that can be realised as torus fibrations over elliptic curves. These moduli
spaces are interesting examples of holomorphic symplectic manifolds whose
geometry is similar to the geometry of Mukai's moduli spaces on K3 and
abelian surfaces. In particular, for certain choices of rank and Chern
classes, the moduli spaces are themselves Kodaira surfaces.
This is joint work with Marian Aprodu and Matei Toma.
Paul Andi Nagy (Greifswald University)
Symplectic forms on Kaehler surfaces
Necessary and sufficient conditions for the
existence of orthogonal almost-Kaehler structures on Kaehler surfaces
will be given. We will explain how
these conditions work on several classes of examples. The relation to
the problem of finding a symplectic form on a Kaehler surface will be
outlined.
Stefan Nemirovski (Steklov Institute)
Universal coverings of strictly pseudoconvex domains
The universal covering of a strictly pseudoconvex domain in a Stein
manifold is completely determined by the local CR-geometry of its
boundary.
I will discuss various results and problems related to this general
principle. This is joint work with Rasul Shafikov.
Although holonomy Spin(9) is only possible for the two
16-dimensional symmetric spaces $\mathbb OP^2$ and
$\mathbb O H^2$, weakened holonomy Spin(9) conditions have
been proposed and studied, in particular by
Th. Friedrich. A basic problem is to have a simple
algebraic formula for the canonical $8$-form
$\Phi_{Spin}(9)}$, similar to the usual definition
of the quaternionic 4-form $\Phi_{\mathrm{Sp}(n)\cdot
\mathrm{Sp}(1)}= \omega_I^2+\omega_J^2+\omega_K^2$, witten
in terms of local compatible almost hypercomplex
structures (I,J,K).
In the talk, a simple formula for
$\Phi_{\mathrm{Spin}(9)}$ is presented, discussing a
family of local almost hypercomplex structures associated
with a Spin(9)-manifold $M^{16}$. Some of these
complex structures, now on model spaces $\R^{16^q}$,
are then used to give an approach through
Spin(9) to the very classical problem of
writing down a maximal system of tangent vector fields on
spheres $S^{N-1} \subset \R^N$. If time permits,
some properties of manifolds equipped with a locally
conformal parallel Spin(9) metric will be also
discussed.
Sönke Rollenske (Johannes Gutenberg-Universität Mainz)
Lagrangian fibrations on hyperkaehler manifolds
Hyperkaehler (also called irreducible holomorphic symplectic) manifolds
form an important class of manifolds with trivial canonical bundle. One
fundamental aspect of their structure theory is the question whether a
given hyperkaehler manifold admits a Lagrangian fibration. I will report
on a joint project with Daniel Greb and Christian Lehn investigating the
following question of Beauville: if a hyperkaehler manifold contains a
complex torus T as a Lagrangian submanifold, does it admit a
(meromorphic) Lagrangian fibration with fibre T? I will describe a
complete positive answer to Beauville's Question for non-algebraic
hyperkaehler manifolds, and give explicit necessary and sufficient
conditions for a positive solution in the general case using the
deformation theory of the pair (X,T).
Yann Rollin (Nantes University)
Stability of extremal metrics under complex deformations
Let $(X,\Omega)$ be a closed polarized complex
manifold, $g$ be an extremal metric on $X$ that represents
the K\"ahler class $\Omega$, and $G$ be a compact
connected subgroup of the isometry group
$Isom(X,g)$. Assume that the Futaki invariant relative to
$G$ is nondegenerate at $g$. Consider a smooth family
$(M\to B)$ of polarized complex deformations of
$(X,\Omega)\simeq (M_0,\Theta_0)$ provided with a
holomorphic action of $G$. Then for every $t\in B$
sufficiently small, there exists an
$h^{1,1}(X)$-dimensional family of extremal K\"ahler
metrics on $M_t$ whose K\"ahler classes are arbitrarily
close to $\Theta_t$.
Song Sun (Imperial College London)
Positivity and Sasakian geometry
We will classify compact Sasaki
manifolds with positive transverse curvature.
This can be viewed as a generalization of
the solution to the Frankel conjecture.
Vladlen Timorin (HSE)
Maps that take lines to conics
We will discuss generalizations of the classical theorem
of Moebius (1827): a one-to-one self-map of a real
projective space that takes all lines to lines is a
projective transformation. E.g. we study sufficiently
smooth local maps taking line segments to parts of
conics. A description of local maps taking line segments
to circle arcs depends non-trivially on the dimension (the
descriptioninvolves classical geometries, quaternionic
Hopf fibrations, representations of Clifford algebras).
For most dimensions, it is still missing.
We will address the problem of understanding the
collapsing of Ricci-flat Kahler metric on abelian fibred
projective Calabi-Yau manifolds. We will then explain an
application of these results to the Strominger-Yau-Zaslow picture
of mirror symmetry for some hyperkahler manifolds.
Joint work with Mark Gross and Yuguang Zhang.
We study the behaviour of locally conformally Kahler (LCK
for short) manifolds under birational transformations. We
show that the blow-up of an LCK manifold X along a
subvariety Y is LCK iff Y is globally conformallly Kahler
(GCK). Using the same methods, we also show
that a twistor space is LCK iff it is GCK. We will also
adress a number of open questions.
This is joint work with L. Ornea and M. Verbitsky.
Home | Venue |
Schedule | Program | Poster | PDF